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Questions and Answers
What is the simplification of the rational expression $\frac{x^2 - 16}{x^2 + 6x + 8}$?
What is the simplification of the rational expression $\frac{x^2 - 16}{x^2 + 6x + 8}$?
c
How many excluded value(s) are there for the expression $\frac{x - 6}{x + 4}$?
How many excluded value(s) are there for the expression $\frac{x - 6}{x + 4}$?
1
What is the excluded value of the expression $\frac{x - 6}{x + 4}$?
What is the excluded value of the expression $\frac{x - 6}{x + 4}$?
-4
What are the excluded value(s) of the rational expression $\frac{x^2 + 2x - 3}{x^2 + 5x + 6}$?
What are the excluded value(s) of the rational expression $\frac{x^2 + 2x - 3}{x^2 + 5x + 6}$?
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Complete the table of the values for the function $f(x) = \frac{1}{x}$ with their corresponding values.
Complete the table of the values for the function $f(x) = \frac{1}{x}$ with their corresponding values.
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What happens to $f(x)$ as $x$ approaches zero from the positive direction?
What happens to $f(x)$ as $x$ approaches zero from the positive direction?
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What happens to $f(x)$ as $x$ approaches zero from the negative direction?
What happens to $f(x)$ as $x$ approaches zero from the negative direction?
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What is the domain of the function $f(x) = \frac{1}{x}$?
What is the domain of the function $f(x) = \frac{1}{x}$?
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Identify the vertical asymptote of the function $f(x) = \frac{x^2 + 1}{3(x - 8)}$.
Identify the vertical asymptote of the function $f(x) = \frac{x^2 + 1}{3(x - 8)}$.
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What are the vertical asymptotes of $f(x) = \frac{10}{x^2 - 1}$?
What are the vertical asymptotes of $f(x) = \frac{10}{x^2 - 1}$?
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The product of wavelength and frequency represented by the equation $k = \text{wavelength} \times \text{frequency}$ yields values such as a = ______.
The product of wavelength and frequency represented by the equation $k = \text{wavelength} \times \text{frequency}$ yields values such as a = ______.
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What is the equation that models the relationship between wavelength and frequency of yellow light?
What is the equation that models the relationship between wavelength and frequency of yellow light?
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For Gamma-Rays, given Frequency = 40 Hz and Wavelength = 300,000, what is the constant?
For Gamma-Rays, given Frequency = 40 Hz and Wavelength = 300,000, what is the constant?
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Identify the vertical asymptote and the hole on the graph of the function $f(x) = \frac{x^2 + x - 6}{x^2 - 6x + 8}$.
Identify the vertical asymptote and the hole on the graph of the function $f(x) = \frac{x^2 + x - 6}{x^2 - 6x + 8}$.
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Find the discontinuities of the function $f(x) = \frac{x^2 + 12x + 27}{x^2 + 4x + 3}$.
Find the discontinuities of the function $f(x) = \frac{x^2 + 12x + 27}{x^2 + 4x + 3}$.
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Where is the vertical asymptote(s) for the given function?
Where is the vertical asymptote(s) for the given function?
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Study Notes
Rational Expressions and Vertical Asymptotes
- Simplification of rational expressions can reveal excluded values. For example, ( \frac{x^2 - 16}{x^2 + 6x + 8} ) and ( \frac{x^2 - x - 6}{x^2 - 3x - 10} ) can be simplified for further analysis.
Excluded Values
- An excluded value in a rational function is a value that makes the denominator equal to zero. For ( \frac{x - 6}{x + 4} ), there is one excluded value: -4.
- The excluded values for ( \frac{x^2 + 2x - 3}{x^2 + 5x + 6} ) are -2 and -3.
Function Behavior Near Zero
- For the function ( f(x) = \frac{1}{x} ), as ( x ) approaches zero from the positive direction, ( f(x) ) becomes larger.
- Conversely, as ( x ) approaches zero from the negative direction, ( f(x) ) increases in the negative direction.
Domain of Functions
- The function ( f(x) = \frac{1}{x} ) has a domain of all real numbers except ( x = 0 ).
Identifying Vertical Asymptotes
- The vertical asymptote for ( f(x) = \frac{x^2 + 1}{3(x - 8)} ) is located at ( x = 8 ).
- For ( f(x) = \frac{10}{x^2 - 1} ), the vertical asymptotes are situated at ( x = 1 ) and ( x = -1 ).
Inverse Variation Relationships
- Inverse variation can be modeled by the relationship between wavelength and frequency, such that the constant ( k ) is equal to the product of wavelength and frequency.
- For yellow light, the modeling constant is found to be 302,400.
- For gamma rays with a frequency of 40 Hz and a wavelength of 300,000, the constant is calculated to be 12,000,000.
Discontinuities
- In the function ( f(x) = \frac{x^2 + 12x + 27}{x^2 + 4x + 3} ), there is a removable discontinuity at the point (-3, -3).
- For the function ( f(x) = \frac{x^2 + x - 6}{x^2 - 6x + 8} ), there is a vertical asymptote at ( x = 4 ) and a hole in the graph at the coordinates (2, -2.5).
Summary of Vertical Asymptotes
- Comprehensive understanding of vertical asymptotes is vital for analyzing rational function behavior, particularly when determining points of discontinuity and behavior around such points.
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Explore the concept of vertical asymptotes with these flashcards on rational functions. Simplify rational expressions and identify excluded values to enhance your understanding. Perfect for students looking to grasp key concepts in algebra.
